Sorry to hear your news. What paper was the 113 score in? My daughter scored 138 (verbal) 133 (non verbal) and 113 (maths) and got in on Headteacher's appeal. We were not aiming for one of the "high" attaining grammars. The school also submitted some work that my daughter had done at home a couple of months before (a project on a subject that really interested her). I was told that the reason the appeal was successful was the high scores in the verbal and non verbal and the good quality of the work submitted. If I were you I would get your child to do some work at home to produce at an appeal - apparently last year's SATs results are also helpful - mid to high 4s or 5s. My daughter's SATs results last year were 5C (English comprehension) 4B (writing) and 4C (maths).
All the very best of luck. Another point - when is your child's birthday?
The 113 was in Maths. The crux of my appeal is that the school is not one that prepares for, or encourages for 11+. In fact 43% of my child's classmates have Special Educational Needs. My argument is that in a less challenging learning environment, the scores would have been a lot better. With the best will in the world, the teachers have more pressing issues than helping five kids out of thirty to pass the 11+ when almost half can barely read.
Age did also count against us. It probably lost us a mark or two when the scores were adjusted.